LHS collapse for a cohomologically trivial normal subgroup

All $E_2^{p,q}$ with $q>0$ are zero, while $E_2^{p,0}=H^p(Q,M^N)$. For $r\ge 2$, a differential out of this bottom row has negative second coordinate, and one into it starts in a zero row. Induction over pages therefore gives $E_2=E_{\mathrm{\infty }}$.

In degree $n$, the only possible quotient is at filtration index $n$. The zero quotients before it imply $F^0{H}^n=\cdots =F^n{H}^n$, and $F^{n+1}{H}^n=0$, so this quotient is the whole target; this is also the lower-edge isomorphism of F2. To identify its map, use the morphism of extensions $(N,G,Q)\to (1,Q,Q)$ given by $N\to 1$, $\pi :G\to Q$, and ${\mathrm{id}}_Q$, together with the $G$-linear inclusion from the inflation of $M^N$ into $M$. By F1's contravariant map-of-extensions naturality, it induces a map from the trivial-kernel LHS sequence for $M^N$ to the given sequence. The source sequence has only its $q=0$ row and its lower edge is the identity on $H^n(Q,M^N)$. The induced map on the target is the usual restriction/coefficient map along $\pi$, namely inflation, while the map on the bottom $E_2$ row is the identity because taking $N$-invariants of $M^N\hookrightarrow M$ recovers $M^N$. Commutativity of the edge square therefore identifies the lower edge above with inflation. At $n=0$ it is $(M^N{)}^Q=M^G$; for $M=0$ or a zero surviving quotient the finite filtration gives zero. The trivial normal group satisfies the vanishing automatically. All comparisons retain F1's precise DC or supplied-data convention.